I don't understand how to get from the first line to the last. What happens to the summation? Please help! An important piece of information given in the question is that the transition probability from state 0 to state 1 is equal to 1. N.B. o(h) is just an error term that goes to 0 when divided by h and a limit as h->0 is taken.

Also we probably use the fact that the probabilities add up to $1$. And we know that $P(\xi_1 = 1| \xi_0 = 0) = 1$. This simplifies to $\lambda h+o(h)$ probably because we can combine terms to get one $o(h)$ term. Put $x = \lambda h$. So (3) is equal to: